Ant Colony Optimization

Operations Research Paper

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Index

Table of Contents Introduction ............................................................................................................................................. 3

Biological Inspiration ............................................................................................................................... 3

Swarm Intelligence .............................................................................................................................. 4

Ant colony algorithm ........................................................................................................................... 4

Pheromone trails ................................................................................................................................. 4

Algorithm ............................................................................................................................................. 5

The optimization technique ..................................................................................................................... 5

The Ant Colony Optimization Metaheuristic ....................................................................................... 6

Application ............................................................................................................................................... 7

Travelling salesman problem ............................................................................................................... 7

Real life example .................................................................................................................................. 8

Advantages: ......................................................................................................................................... 8

Disadvantages ...................................................................................................................................... 9

Theoretical Results .................................................................................................................................. 9

Outlook and Conclusions ....................................................................................................................... 10

References ............................................................................................................................................. 10

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Introduction Ant colony optimization (ACO) takes inspiration from the foraging behavior of some ant species.

This approach is derived from Swarm intelligence. Swarm intelligence is a relatively new approach to problem solving that takes inspiration from the social behaviors of insects and of other animals. The study behavior of ants had been the most successful till date. Thus Ant colony optimization is widely applied in computer science and operations research. It is a probabilistic technique for solving computational problems which can be reduced to finding good paths.

These are a set of algorithms which come under swarm intelligence methods and metaheuristics optimizations. Initially it was proposed by Marco Dorigo in 1992. Since then ACO has attracted the attention of increasing numbers of researchers and many successful applications are now available. Moreover, a substantial corpus of theoretical results is becoming available that provides useful guidelines to researchers and practitioners in further applications of ACO.

In this study we will first, deal with the biological inspiration of ant colony optimization algorithms. We show how this biological inspiration can be transferred into an algorithm for discrete optimization. Then, we study the optimization techniques, algorithm and theoretical results.

Biological Inspiration Some species react to significant stimuli, and effects of these reactions can act as new

significant stimuli for both the insect that produced them and for the other insects in the colony. Stigmergy is particular type of communication in which workers are stimulated by the performance they have achieved. The two main characteristics of stigmergy that differentiate it from other forms of communication are the following.

1) Stigmergy is an indirect, non-symbolic form of communication mediated by the environment: insects exchange information by modifying their environment; and

2) Stigmergic information is local: it can only be accessed by those insects that visit the locus in which it was released (or its immediate neighborhood).

In many ant species, ants walking to and from a food source deposit on the ground a substance called pheromone. Other ants perceive the presence of pheromone and tend to follow paths where pheromone concentration is higher. Through this mechanism, ants are able to transport food to their nest in a remarkably effective way.

In an experiment known as the double bridge experiment, the nest of a colony of Argentine ants was connected to a food source by two bridges of equal lengths. In such a setting, ants start to explore the surroundings of the nest and eventually reach the food source. Along their path between food source

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and nest, Argentine ants deposit pheromone. Initially, each ant randomly chooses one of the two bridges. However, due to random fluctuations, after some time one of the two bridges presents a higher concentration of pheromone than the other and, therefore, attracts more ants. This brings a further amount of pheromone on that bridge making it more attractive with the result that after some time the whole colony converges toward the use of the same bridge. This colony-level behavior, based on autocatalysis, that is, on the exploitation of positive feedback, can be used by ants to find the shortest path between a food source and their nest.

Ant colony optimization is a technique for optimization that was introduced in the early 1990s. The inspiring source of ant colony optimization is the foraging behavior of real ant colonies.

Swarm Intelligence It is a collective system capable of accomplishing difficult tasks in dynamic and varied

environments without any external guidance or control and with no central coordination. It is achieved by a collective performance and cannot normally be achieved by an individual acting alone. It constitutes a natural model particularly suited to different kind of problem solving. Swarm intelligence has inspired to create some highly successful optimization algorithms. One of those algorithms is ant colony algorithm. It is a way to solve optimization problems based on the behavior of ants searching for food.

Ant colony algorithm The principle is that the trace (stigmergy) left in the environment by an action stimulates the

performance of a next action, by the same or a different agent. Individuals leave markers or messages these dont solve the problem in themselves, but they affect other individuals in a way that helps them solve the problem.

Pheromone trails One ant tends to follow strong concentration of pheromone caused by repeated passes of ants;

a pheromone trail is then formed from nest to food source, so in intersections between several trails an ant moves with high probability following the highest pheromone level.

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Individual ants lay pheromone trails while travelling from the nest, to the nest or possibly in both directions. The pheromone trail gradually evaporates over time. But pheromone trail strength accumulates with multiple ants using path.

Example of ant colony optimization:

Algorithm Ants are agents that move along between nodes in a graph. They choose where to go based on

pheromone strength (and maybe other things). An ants path represents a specific candidate solution. When an ant has finished a solution, pheromone is laid on its path, according to quality of solution. This pheromone trail affects behaviour of other ants by `stigmergy

The optimization technique It was put forward by Deneubourg and team after getting inspired by the foraging behavior of ants. The algorithm works by going through a number of iterations of all the possible solutions and every time updating the solution so that the next iteration leverages on the learning. The leveraging is done using the environment as a means of communication. This original idea was proposed in early 90s but later many algorithms used this algorithm as the base and developed new forms of it.

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The Ant Colony Optimization Metaheuristic Metaheuristic: A metaheuristic is a set of algorithmic concepts that can be used to define heuristic methods applicable to a wide set of different problems. In simple terms it is a general purpose algorithm that can be applied to different problems. The problem can be defined as follows.

As the above definition explains, there are 3 main aspects to define the problem.

1. A set of decision variable 2. All the possible constrains the solution should follow 3. A function that defines all the possible solutions

The best solution of the above defined problem is the one which give the most optimum solution. This optimum value is defined by a variable which is analogues to the pheromone value in the ant case. These values are calculated for each possible solution which are the outcome of the multiple iterations of the ant, and compared against each other considering the requirements and the optimum solution is chosen.

The ACO metaheuristic is shown in Algorithm 1.

After initialization, the metaheuristic iterates over three phases explained below,

Construct Ant Solutions:

A set of m artificial ants constructs solutions from elements of a finite set of available solution components C = {cij }, i = 1, . . . , n, j = 1, . . . , |Di | . A solution construction starts from an empty partial solution sp = . At each construction step, the partial solution sp is extended by adding a feasible solution component from the set N(s p) C, which is defined as the set of components that can be

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added to the current partial solution s p without violating any of the constraints. The choice of a solution component from N(s p) is guided by a stochastic mechanism, which is biased by the variable that depicts the pheromone associated with each of the elements of N(s p). The rule for the stochastic choice of solution components vary across different ACO algorithms but in all of them, it is inspired by the model of the behavior of real ants given in Equation 1.

Apply Local Search:

Once solutions have been constructed, and before updating the pheromone, it is common to improve the solutions obtained by the ants through a local search. This phase, which is highly problem-specific, is optional although it is usually included in state-of-the-art ACO algorithms.

Update Pheromones:

The aim of the pheromone update is to increase the pheromone values associated with good or promising solutions, and to decrease those that are associated with bad ones. Usually, this is achieved (i) by decreasing all the pheromone values through pheromone evaporation, and (ii) by increasing the pheromone levels associated with a chosen set of good solutions.

Application

Travelling salesman problem Given a list of cities and the distances between each pair of cities, ACI is used to find the shortest

possible route that visits each city exactly once and returns to the origin city. The ant algorithm is shown below:

1. Ant is placed at a random node as seen in the diagram at node B. 2. The ant decides where to go from that node, based on probabilities calculated from: pheromone

strengths, next-hop distances. Suppose this one chooses BC. The ant is now at C, and has a tour memory = {B, C} so he cannot visit B or C again. Again, he decides next hop(from those allowed) based on pheromone strength and distance; suppose he chooses CD

3. The ant is now at D, and has a `tour memory = {B, C, D}. There is only one place he can go now: 4. So, he has nearly finished his tour, having gone over the links: BC, CD, and DA. 5. So, he has nearly finished his tour, having gone over the links: BC, CD, and DA. AB is added to

complete the round trip. Now, pheromone on the tour is increased, in line with the fitness of that tour.

6. Next, pheromone everywhere is decreased a little, to model decay of trail strength over time. 7. We start again, with another ant in a random position.

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1. 2. 3. 4.

5. 6.

7. AB =10, AC =10, AD =30, BC =40, CD =20

Real life example We place some salesman at each city. Each salesman then does this:

It makes a complete tour of the cities, coming back to its starting city, using a transition rule to decide which links to follow. By this rule, he chooses each next-city at random, but biased partly by the pheromone levels existing at each path, and biased partly by heuristic information.

When all salesmen have completed their tours. Global Pheromone Updating occurs.

The current pheromone levels on all links are reduced Then we go back to 1 and repeat the whole process many times, until we reach a termination

criterion.

Advantages: Positive Feedback accounts for rapid discovery of good solutions Distributed computation avoids premature convergence

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The greedy heuristic helps find acceptable solution in the early solution in the early stages of the search process.

The collective interaction of a population of agents.

Disadvantages Slower convergence than other Heuristics Performed poorly for TSP problems larger than 75 cities. No centralized processor to guide the system towards good solutions.

Theoretical Results Experimental work in this regard has shown that successful algorithms can be derived from ACO. Certain theoretical foundations have also been conceptualized based on the study by researchers. The work highlights the following questions while dealing with metaheuristics concerns:

Will an optimal solution be derived using the given ACO algorithm?

Some of the first convergence proofs was provided in the form of Graph Based Ant System (GBAS). The probability found out under this model comes out to be 1-. However, the above algorithm is rather peculiar and does not extend to other algorithms generally adopted in applications. Another observation is that these convergence results do not predict the timeline or the time taken to find the optimal solution. Recently, Gutjahr presented models to predict these timelines.

This research has led to opening of new channels:

1) Link between ACO and optimal control and reinforcement theory. 2) Link between ACO and probabilistic learning algorithms.

Also a more wholesome Model Based Search (MBS) algorithm has also been proposed which is claimed to improve understanding of the ACO.

Convergence proofs do not generally provide implementation guidelines to researchers. In this regard, research efforts that aim for higher understanding propose better solutions. Also, first order deception has also been found in ACO algorithms along with second order deception.

Some of the ACO applications are listed below:

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Outlook and Conclusions The first ACO algorithm which was inspired by ant colonies was introduced almost 15 years ago.

Despite of inviting huge flack during the initial years of its inception the technique presents most promising techniques to difficult optimization problems. As seen in recent years, ACO and its variations will be a major research channel in the future.

References https://www.ics.uci.edu/~welling/teaching/271fall09/antcolonyopt.pdf rain.ifmo.ru/~chivdan/presentations www.macs.hw.ac.uk/~dwcorne/Teaching Dorigo M, Optimization, learning and natural algorithms. PhD thesis, Dipartimento di

Elettronica, Politecnico di Milano, Italy, 1992 [in Italian] Wikipedia https://www.ics.uci.edu/~welling/teaching/271fall09 code.ulb.ac.be/dbfiles/ Ant Colony Optimization for Feature Selection in Software Product Lines by WANG Ying-lin1,2,

PANG Jin-wei2 mitpress.mit.edu/books/ant-colony-optimization